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A325933 Partial sums of A325931.

%I A325933 #20 Jan 09 2021 22:45:14
%S A325933 0,1,2,3,4,3,4,5,4,5,4,5,6,5,6,5,6,5,6,5,6,7,6,7,6,7,6,7,6,7,6,7,6,7,
%T A325933 6,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,9,8,9,8,9,8,9,
%U A325933 8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9,8,9
%N A325933 Partial sums of A325931.
%C A325933 This sequence is the "tier profile" of A076042, much as A064289 provides a similar profile of Recamán's Sequence (A005132). In each case, the base sequence jumps around between adjacent "tiers", each of which changes value relatively slowly (compared to the difference between entries in different tiers). As in Recamán's sequence, each tier rises for a while, then crests, and finally falls close to zero, at which point that tier becomes extinct. The tier structure is salient when viewing the graph of such a sequence.
%C A325933 This sequence is less chaotic than A064289, since in A076042 only two tiers are active at time, and the sequence alternates between the two until the lower tier stops, at which point a double increase inaugurates a new tier.
%H A325933 Alois P. Heinz, <a href="/A325933/b325933.txt">Table of n, a(n) for n = 0..65536</a>
%F A325933 a(0) = 0; a(n) = a(n-1) + A325931(n).
%e A325933 a(20) = a(19) + A325931(20) = 5 + 1 = 6.
%p A325933 b:= proc(n) option remember; `if`(n=0, 0, (t->
%p A325933       t+`if`(t<n^2, 1, -1)*n^2)(b(n-1)))
%p A325933     end:
%p A325933 a:= proc(n) option remember; `if`(n=0, 0,
%p A325933       a(n-1)+signum(b(n)-b(n-1)))
%p A325933     end:
%p A325933 seq(a(n), n=0..105);  # _Alois P. Heinz_, Sep 08 2019
%t A325933 b[n_] := b[n] = If[n == 0, 0, b[n - 1] + If[b[n - 1] < n^2, n^2, -n^2]];
%t A325933 b /@ Range[0, 105] // Differences // Sign // Accumulate // Prepend[#, 0]& (* _Jean-François Alcover_, Nov 30 2020 *)
%Y A325933 Cf. A076042, A325931.
%K A325933 nonn,easy
%O A325933 0,3
%A A325933 _Allan C. Wechsler_, Sep 08 2019